A key issue in Global Navigation Satellite Systems (GNSS), including the original Global Positioning System (GPS), is the isolation and estimation of the integer ambiguity of the carrier phase measurements. Achieving integer ambiguity resolution of undifferenced carrier phase measurements has been a significant topic of interest for the geodetic and navigation communities for many years.
The carrier phase signals of any GNSS are approximately two orders of magnitude more precise than the primary pseudorange signals the systems provide. However, measurements of the carrier phases are ambiguous relative to those of the pseudoranges by an unknown number of integer cycles. Resolution of these ambiguities provides centimeter to millimeter-level pseudorange measurements compared to the decimeter to meter level of the inherent pseudoranges. The improvement in measurement precision is directly carried into the parameters estimated from the measurements.
The ambiguities are currently only resolved as integers in so-called double difference processing whereby dual-pairs of measurements (made for example from two receivers to the same two satellites) are differenced to produce one measurement. The differencing is carried out primarily to remove common transmitter and receiver biases contained in the measurements. The biases consist primarily of the oscillator-induced time delays of both the satellite and the receiver. The existence of these biases is why the measurements are ‘pseudoranges’ and not just ‘ranges’.
The primary disadvantage of double differencing is the requirement to have at least two receivers, even for a single user who only requires his own location. This essentially turns point positioning into baseline, or relative, positioning. This technique can be very limited in baseline length if such error sources as orbits and ionosphere are required to cancel out.
As an alternative to double differencing, it is possible to process undifferenced measurements and estimate the biases explicitly. It can be shown that the two solutions are mathematically identical under certain circumstances. At the same time however, it is not possible to explicitly isolate the integer nature of the ambiguities due to their exact linear correlation with time delays due to the oscillators and other hardware. The higher precision of the carrier phase can still be accessed by estimating a random constant bias in place of the ambiguity; however, such a parameter requires an extended convergence period.
The processing of undifferenced pseudorange and carrier phase observables from a single receiver is referred to as Precise Point Positioning (PPP). PPP returns in effect to the first principles of GPS, where the focus is again placed on a single receiver. The main challenge with PPP is the significant convergence period required before a suitable solution precision is achieved. This convergence period is the most significant factor limiting wider adoption of PPP. If the ambiguity could be isolated and estimated as an integer value then, in principle, the integer nature represents more information that could be exploited to accelerate convergence.
Accordingly, integer ambiguity resolution of undifferenced carrier phase observables has been an elusive goal in GPS processing, largely since the advent of the PPP method. Some recent advances in isolating integer ambiguities have been made with techniques that use single differences and undifferenced observables. However, it is not clear that all aspects of the problem have been addressed, particularly with respect to time-varying code biases that are not explicitly accounted for in those techniques.
The term ‘code biases’ generally refers to unmodelled common-mode errors of the pseudoranges, usually considered to be hardware or local environment delays, that are either constant or believed to vary in a band-limited, quasi-random, manner. There appears to be general acceptance in the timing community that these biases are the cause of the so-called ‘day-boundary clock jumps’ highlighted by the time scale of the International GNSS Service (IGS). The term ‘phase biases’ refers to corresponding delays of the carrier phases.
It is stated that the limiting factor in ambiguity resolution using undifferenced GPS observables is the presence of both code and phase biases in the estimates of the ambiguities. As parameterised in the “standard model” of undifferenced pseudoranges and carrier phases, the datum for the station and satellite clock parameters is provided by the pseudoranges. The consequence of this is that the estimated ambiguities contain the time-constant portions of both code and phase biases.
The standard GPS dual-frequency pseudorange (code) and carrier phase (phase) observation equations are typically written in the form:
                              P          1                =                  ρ          +          T          +          I          +                      c            ⁡                          (                                                dt                  r                                -                                  dt                  s                                            )                                +                      b                          P              ⁢                                                          ⁢              1                        r                    -                      b                          P              ⁢                                                          ⁢              1                        s                    +                      ɛ                          P              ⁢                                                          ⁢              1                                                                        P          2                =                  ρ          +          T          +                                    q              2                        ⁢            I                    +                      c            ⁡                          (                                                dt                  r                                -                                  dt                  s                                            )                                +                      b                          P              ⁢                                                          ⁢              2                        r                    -                      b                          P              ⁢                                                          ⁢              2                        s                    +                      ɛ                          P              ⁢                                                          ⁢              2                                                                                    λ            1                    ⁡                      (                                          Φ                1                            +                              N                1                                      )                          =                                          L          1                =                  ρ          +          T          -          I          +                      c            ⁡                          (                                                dt                  r                                -                                  dt                  s                                            )                                +                      b                          L              ⁢                                                          ⁢              1                        r                    -                      b                          L              ⁢                                                          ⁢              1                        s                    +                      ɛ                          L              ⁢                                                          ⁢              1                                                                                    λ            2                    ⁡                      (                                          Φ                2                            +                              N                2                                      )                          =                                          L          2                =                  ρ          +          T          -                                    q              2                        ⁢            I                    +                      c            ⁡                          (                                                dt                  r                                -                                  dt                  s                                            )                                +                      b                          L              ⁢                                                          ⁢              2                        r                    -                      b                          L              ⁢                                                          ⁢              2                        s                    +                      ɛ                          L              ⁢                                                          ⁢              2                                          where Pi is a pseudorange measurement made at frequency i and Φi is a carrier phase measurement made at frequency i. We write the integer ambiguity Ni on the left side to show how it converts the ambiguous phase measurement Φi into a precise pseudorange Li. The factor q represents the ratio of the primary and secondary GPS frequencies, c is the vacuum speed of light and λi is the frequency-dependent wavelength of the carrier phase measurements. Of the geometric parameters, ρ represents the geometric range between transmitter and receiver antennas, T is the range delay caused by signal propagation through the lower atmosphere (predominantly the Troposphere), and I is the range delay and apparent phase advance on the primary frequency caused by signal propagation through the upper atmosphere (predominantly the Ionosphere). The remaining, non-geometric, parameters are the oscillator or ‘clock’ errors for both the transmitter and receiver (dts and dtr respectively), and common-oscillator hardware biases (b**) for each observation. Unmodelled random or quasi-random errors are represented by ε*. 
The usual practice when processing dual-frequency measurements is to take advantage of the frequency-dependence of the ionospheric delay (to first order) and linearly combine two pseudoranges and two carrier phases to produce ionosphere-free observables:P3=ρ+T+c(dtr−dts)+bP3r−bP3s+εP3 L3=ρ+T+c(dtr−dts)+bL3r−bL3s−λ3N3εL3 where:
                                                        P              3                        =                                          (                                                                            77                      2                                        ⁢                                          P                      1                                                        -                                                            60                      2                                        ⁢                                          P                      2                                                                      )                                            (                                                      77                    2                                    -                                      60                    2                                                  )                                              ,                                                  L            3                    =                      ⁢            (                                    77            2                    ⁢                      L            1                          -                              60            2                    ⁢                      L            2                              )              (                        77          2                -                  60          2                    )      and the ionosphere-free ambiguity combination N3=77N1−60N2 is placed on the right-hand side to indicate that it is now treated as a parameter to be estimated from the data.
As they stand, these equations are over-parameterised, and any system of normal equations derived from them for the purposes of a least squares solution will be singular. There are two causes of deficiency. The first is that the clock errors are inherently differenced and cannot be uniquely separated. This is overcome in a network solution by fixing one of the station clocks, and in a single-receiver solution by fixing the satellite clocks. The remaining singularity is due to the presence of the hardware biases and their identical functional behavior with the associated clock parameters. Both these types of parameter represent common-mode time delays and having constant partial derivatives are not uniquely separable.
To explicitly deal with this singularity, equivalent equations can be written with clock and code bias parameters combined. At the same time, due to the uniquely ambiguous nature of the carrier phase, the combined clock and bias parameter can be carried over to the phase observable:P3=ρ+T+c(dtP3r−dtP3s)+εP3 L3=ρ+T+c(dtP3r−dtP3s)+AP3+εL3 where c·dtP3*=c·dt*+bP3*, and compensating code biases plus the phase biases and the ambiguity are combined into one parameter:AP3=bL3r−bP3r−bL3+bP3s−λ3N3 which is sometimes referred to as the ‘float ambiguity’ because it is not integer valued. The justification for grouping these parameters is that they are all functionally identical (constant partial derivatives) and as a random bias, the integer ambiguity cannot be independently predicted a-priori.
These equations will be referenced as the standard model for dual-frequency undifferenced processing, despite being in what may appear to be a non-standard form. These equations correctly represent the combined effect of common-mode code-biases, common-observation clocks and random bias ambiguities. The net effect is that the ambiguity parameter of the standard model contains both phase and code time-constant biases.
It is re-stated that the standard model of undifferenced ionosphere-free observables is sub-optimal, in that the estimated ambiguities contain the constant code as well as phase biases. Should the code or phase biases also vary over time, then the standard model is even less accurate, and any such variations must be accommodated by the other estimated parameters. For the standard model this will be primarily the clocks and ambiguities.
Prior systems have not addressed this problem. For example, U.S. Pat. Nos. 5,621,646 and 5,828,336 both describe systems to compute GPS user corrections from wide-area ground networks. Both refer to using ionosphere-free pseudoranges explicitly smoothed with the carrier phase measurements, or averaged pseudoranges as estimates for the carrier ambiguity. The latter patent describes a process to explicitly estimate the ionosphere through which the measurements pass, wherein the existence of “instrumental biases” in both the receiver and satellite transmitter is acknowledged. The differing nature of code biases versus phase biases is not acknowledged and neither is the impact on standard model processing for positioning, etc. U.S. Pat. No. 5,963,167 (which is a development of U.S. Pat. No. 5,828,336) describes a more comprehensive analysis package for processing GPS observations. This patent refers briefly to the fixing of double-difference ambiguities in a network solution, but without significant detail and not for an isolated user.
U.S. Pat. No. 6,697,736 B2 relates to a positioning and navigation method and system combining GPS with an Inertial Navigation System. The objective of this patent is to provide a positioning and navigation method and system, in which the satellite signal carrier phase measurements, as well as the pseudoranges of the Global Positioning System are used in a Kalman filter, so as to improve the accuracy of the integrated positioning solution. It is claimed that carrier phase ambiguities can be fixed, but the existence and effect of code and phase biases in the underlying mathematical model is not recognised.